Reciprocal Prime Magic Squares - Joel Haddley

24th November 2012

It is well known that rational numbers are numbers whose decimal expansions either terminate or repeat indefinitely. If \(p\) is a prime number, it turns out that the numbers in its decimal expansion repeat with period dividing \(p-1\). We will explain why this happens, state under what condition the period is exactly \(p-1\), and explain how all of this relates to magic squares such as:

1
4
2
8
5
7
2
8
5
7
1
4
4
2
8
5
7
1
5
7
1
4
2
8
7
1
4
2
8
5
8
5
7
1
4
2